First Open the SharePoint site and Navigate to the page where you want to add the webpart. Then from the Site Actions -> Edit Page. or also you can directly click on the Edit icon to edit the page as shown in the figure below:

Step-2:

The above step will open the page in Edit Mode. Now From the RIbbon from the Editing Tools section click on Insert and then click on WebPart as shown in the figure below:

Step-3:

The above step will show you all the webparts categorywise. From there select any webpart. Here I am selecting a Content Editor webpart from the Media and Content Category. After that click on Add as shown in the figure below:

Once you will Click on Add the webpart will be placed in the page.

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